1000 Solved Problems in Modern Physics

Transkript

1 1000 Solved Problems in Modern Physics

2 Ahmad A. Kamal 1000 Solved Problems in Modern Physics 123

3 Dr. Ahmad A. Kamal 425 Silversprings Lane Murphy, TX 75094, USA ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: estudio Calamar Steinen Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 Dedicated to my parents

5 Preface This book is targeted mainly to the undergraduate students of USA, UK and other European countries, and the M.Sc of Asian countries, but will be found useful for the graduate students, Graduate Record Examination (GRE), Teachers and Tutors. This is a by-product of lectures given at the Osmania University, University of Ottawa and University of Tebrez over several years, and is intended to assist the students in their assignments and examinations. The book covers a wide spectrum of disciplines in Modern Physics, and is mainly based on the actual examination papers of UK and the Indian Universities. The selected problems display a large variety and conform to syllabi which are currently being used in various countries. The book is divided into ten chapters. Each chapter begins with basic concepts containing a set of formulae and explanatory notes for quick reference, followed by a number of problems and their detailed solutions. The problems are judiciously selected and are arranged section-wise. The solutions are neither pedantic nor terse. The approach is straight forward and step-bystep solutions are elaborately provided. More importantly the relevant formulas used for solving the problems can be located in the beginning of each chapter. There are approximately 150 line diagrams for illustration. Basic quantum mechanics, elementary calculus, vector calculus and Algebra are the pre-requisites. The areas of Nuclear and Particle physics are emphasized as revolutionary developments have taken place both on the experimental and theoretical fronts in recent years. No book on problems can claim to exhaust the variety in the limited space. An attempt is made to include the important types of problems at the undergraduate level. Chapter 1 is devoted to the methods of Mathematical physics and covers such topics which are relevant to subsequent chapters. Detailed solutions are given to problems under Vector Calculus, Fourier series and Fourier transforms, Gamma and Beta functions, Matrix Algebra, Taylor and Maclaurean series, Integration, Ordinary differential equations, Calculus of variation Laplace transforms, Special functions such as Hermite, Legendre, Bessel and Laguerre functions, complex variables, statistical distributions such as Binomial, Poisson, Normal and interval distributions and numerical integration. Chapters 2 and 3 focus on quantum physics. Chapter 2 is basically concerned with the old quantum theory. Problems are solved under the topics of debroglie vii

6 viii Preface waves, Bohr s theory of hydrogen atom and hydrogen-like atoms, positronium and mesic atoms, X-rays production and spectra, Moseley s law and Duan Hunt law, spectroscopy of atoms and molecules, which include various quantum numbers and selection rules, and optical Doppler effect. Chapter 3 is concerned with the quantum mechanics of Schrodinger and Hesenberg. Problems are solved on the topics of normalization and orthogonality of wave functions, the separation of Schrodinger s equation into radial and angular parts, 1-D potential wells and barriers, 3-D potential wells, Simple harmonic oscillator, Hydrogen-atom, spatial and momentum distribution of electron, Angular momentum, Clebsch Gordon coefficients ladder operators, approximate methods, scattering theory-phase-shift analysis and Ramsuer effect, the Born approximation. Chapter 4 deals with problems on Thermo dynamic relations and their applications such a specific heats of gases, Joule Thompson effect, Clausius Clapeyron equation and Vander waal s equation, the statistical distributions of Boltzmann and Fermi distributions, the distribution of rotational and vibrational states of gas molecules, the Black body radiation, the solar constant, the Planck s law and Wein s law. Chapter 5 is basically related to Solid State physics and material science. Problems are covered under the headings, crystal structure, Lattice constant, Electrical properties of crystals, Madelung constant, Fermi energy in metals, drift velocity, the Hall effect, the Debye temperature, the intrinsic and extrinsic semiconductors, the junction diode, the superconductor and the BCS theory, and the Josephson effect. Chapter 6 deals with the special theory of Relativity. Problems are solved under Lorentz transformations of length, time, velocity, momentum and energy, the invariance of four-momentum vector, transformation of angles and Doppler effect and threshold of particle production. Chapters 7 and 8 are concerned with problems in low energy Nuclear physics. Chapter 7 covers the interactions of charged particles with matter which include kinematics of collisions, Rutherford Scattering, Ionization, Range and Straggiling, Interactions of radiation with matter which include Compton scattering, photoelectric effect, pair production and nuclear resonance fluorescence, general radioactivity which includes problems on chain decays, age of earth, Carbon dating, alpha decay, Beta decay and gamma decay. Chapter 8 is devoted to the static properties of nuclei such as nuclear masses, nuclear spin and parity, magnetic moments and quadrupole moments, the Nuclear models, the Fermi gas model, the shell model, the liquid drop model and the optical model, problems on fission and fusion and Nuclear Reactors. Chapters 9 and 10 are concerned with high energy physics. Chapter 9 covers the problems on natural units, production, interactions and decays of high energy unstable particles, various types of detectors such as ionization chambers, proprortional and G.M. counters, Accelerators which include Betatron, Cyclotron, Synchro- Cyclotron, proton and electron Synchrotron, Linear accelerator and Colliders. Chapter 10 deals with the static and dynamic properties of elementary particles and resonances, their classification from the point of view of the Fermi Dirac and Bose Einstein statistics as well as the three types of interactions, strong, Electro-

7 Preface ix magnetic and weak, the conservation laws applicable to the three types of interactions, Gell-mann s formula, the properties of quarks and classification into supermultiplets, the types of weak decays and Cabibbo s theory, the neutrino oscillations, Electro Weak interaction, the heavy bosons and the Standard model. Acknowledgements It is a pleasure to thank Javid for the bulk of typing and suggestions and Maryam for proof reading. I am indebted to Muniba for the line drawings, to Suraiya, Maqsood and Zehra for typing and editing. I am grateful to the Universities of UK and India for permitting me to use their question papers cited in the text to CERN photo service for the cover page to McGraw-Hill and Co: for a couple of diagrams from Quantum Mechanics, L.I. Schiff, 1955, to Cambridge University Press for using some valuable information from Introduction to High Energy Physics, D.H. Perkins and to Ginn and Co: and Pearson and Co: for access to Differential and Integral Calculus, William A. Granville, My thanks are due to Springer-Verlag, in particular Claus Ascheron, Adelheid Duhm and Elke Sauer for constant encouragement. Murphy, Texas February 2010 Ahmad A. Kamal

8 Contents 1 Mathematical Physics Basic Concepts and Formulae Problems VectorCalculus FourierSeriesandFourierTransforms Gamma and Beta Functions Matrix Algebra MaximaandMinima Series Integration OrdinaryDifferentialEquations LaplaceTransforms Special Functions ComplexVariables CalculusofVariation StatisticalDistributions NumericalIntegration Solutions VectorCalculus FourierSeriesandFourierTransforms Gamma and Beta Functions Matrix Algebra MaximaandMinima Series Integration OrdinaryDifferentialEquations LaplaceTransforms Special Functions ComplexVariables CalculusofVariation StatisticalDistribution NumericalIntegration xi

9 xii Contents 2 Quantum Mechanics I Basic Concepts and Formulae Problems debrogliewaves Hydrogen Atom X-rays Spin and μ and Quantum Numbers Stern Gerlah s Experiment Spectroscopy Molecules Commutators Uncertainty Principle Solutions debrogliewaves Hydrogen Atom X-rays Spin and μ and Quantum Numbers Stern Gerlah s Experiment Spectroscopy Molecules Commutators Uncertainty Principle Quantum Mechanics II Basic Concepts and Formulae Problems Wave Function Schrodinger Equation PotentialWellsandBarriers Simple Harmonic Oscillator Hydrogen Atom Angular Momentum Approximate Methods Scattering (Phase-Shift Analysis) Scattering(BornApproximation) Solutions Wave Function Schrodinger Equation PotentialWellsandBarriers Simple Harmonic Oscillator Hydrogen Atom Angular Momentum Approximate Methods Scattering (Phase Shift Analysis) Scattering(BornApproximation)...240

10 Contents xiii 4 Thermodynamics and Statistical Physics Basic Concepts and Formulae Problems Kinetic Theory of Gases Maxwell s Thermodynamic Relations StatisticalDistributions Blackbody Radiation Solutions Kinetic Theory of Gases Maxwell s Thermodynamic Relations StatisticalDistributions Blackbody Radiation Solid State Physics Basic Concepts and Formulae Problems CrystalStructure Crystal Properties Metals Semiconductors Superconductor Solutions CrystalStructure Crystal Properties Metals Semiconductors Superconductor Special Theory of Relativity Basic Concepts and Formulae Problems LorentzTransformations Length, Time, Velocity Mass,Momentum,Energy Invariance Principle Transformation of Angles and Doppler Effect Threshold of Particle Production Solutions LorentzTransformations Length, Time, Velocity Mass,Momentum,Energy Invariance Principle Transformation of Angles and Doppler Effect Threshold of Particle Production

11 xiv Contents 7 Nuclear Physics I Basic Concepts and Formulae Problems KinematicsofScattering RutherfordScattering Ionization, Range and Straggling ComptonScattering Photoelectric Effect Pair Production CerenkovRadiation Nuclear Resonance Radioactivity (General) Alpha-Decay Beta-Decay Solutions KinematicsofScattering RutherfordScattering Ionization, Range and Straggling ComptonScattering Photoelectric Effect Pair Production CerenkovRadiation Nuclear Resonance Radioactivity (General) Alpha-Decay Beta-Decay Nuclear Physics II Basic Concepts and Formulae Problems AtomicMassesandRadii ElectricPotentialandEnergy Nuclear Spin and Magnetic Moment Electric Quadrupole Moment Nuclear Stability Fermi Gas Model Shell Model Liquid Drop Model Optical Model Nuclear Reactions (General) Cross-sections Nuclear Reactions via Compound Nucleus DirectReactions Fission and Nuclear Reactors Fusion...447

12 Contents xv 8.3 Solutions AtomicMassesandRadii ElectricPotentialandEnergy Nuclear Spin and Magnetic Moment Electric Quadrupole Moment Nuclear Stability Fermi Gas Model Shell Model Liquid Drop Model Optical Model Nuclear Reactions (General) Cross-sections Nuclear Reactions via Compound Nucleus DirectReactions Fission and Nuclear Reactors Fusion Particle Physics I Basic Concepts and Formulae Problems SystemofUnits Production Interaction Decay Ionization Chamber, GM Counter and Proportional Counters Scintillation Counter Cerenkov Counter SolidStateDetector Emulsions Motion of Charged Particles in Magnetic Field Betatron Cyclotron Synchrotron Linear Accelerator Colliders Solutions SystemofUnits Production Interaction Decay Ionization Chamber, GM Counter and Proportional Counters Scintillation Counter...515

13 xvi Contents Cerenkov Counter SolidStateDetector Emulsions Motion of Charged Particles in Magnetic Field Betatron Cyclotron Synchrotron Linear Accelerator Colliders Particle Physics II Basic Concepts and Formulae Problems ConservationLaws StrongInteractions Quarks Electromagnetic Interactions WeakInteractions Electro-WeakInteractions FeynmanDiagrams Solutions ConservationLaws StrongInteractions Quarks Electromagnetic Interactions WeakInteractions Electro-weakInteractions FeynmanDiagrams Appendix: Problem Index Index...633

14 Chapter 1 Mathematical Physics 1.1 Basic Concepts and Formulae Vector calculus Angle between two vectors, cos θ = A.B A B Condition for coplanarity of vectors, A.B C = 0 Del = x î + y ĵ + z ˆk Gradient φ = ( φ x î + φ y ĵ + φ ) z ˆk Divergence If V (x, y, z) = V 1 î + V 2 ĵ + V 3 ˆk, be a differentiable vector field, then.v = x V 1 + y V 2 + z V 3 Laplacian 2 = 2 x y = 1 r 2 r ( r 2 r 2 = 2 r r r r 2 θ z 2 (Cartesian coordinates x, y, z) z 2 ) + 1 ( sin θ ) 1 + r 2 sin θ θ θ r 2 sin 2 θ 2 Φ 2 (Spherical coordinates r,θ,φ) (Cylindrical coordinates r,θ,z) Line integrals (a) C φ dr 1

15 2 1 Mathematical Physics (b) C A. dr (c) C A dr where φ is a scalar, A is a vector and r = xî + y ĵ + z ˆk, is the positive vector. Stoke s theorem C A. dr = ( A). n ds = ( A). ds S The line integral of the tangential component of a vector A taken around a simple closed curve C is equal to the surface integral of the normal component of the curl of A taken over any surface S having C as its boundary. Divergence theorem (Gauss theorem) V. A dv = S S A.ˆn ds The volume integral is reduced to the surface integral. Fourier series Any single-valued periodic function whatever can be expressed as a summation of simple harmonic terms having frequencies which are multiples of that of the given function. Let f (x) be defined in the interval ( π, π) and assume that f (x) has the period 2π, i.e. f (x + 2π) = f (x). The Fourier series or Fourier expansion corresponding to f (x) is defined as f (x) = 1 2 a 0 + (a 0 cos nx + b n sin nx) (1.1) n=1 where the Fourier coefficient a n and b n are a n = 1 π b n = 1 π π π π π f (x) cos nx dx (1.2) f (x)sinnx dx (1.3) where n = 0, 1, 2,... If f (x) is defined in the interval ( L, L), with the period 2L, the Fourier series is defined as f (x) = 1 2 a 0 + (a n cos(nπ x/l) + b n sin(nπ x/l)) (1.4) n=1 where the Fourier coefficients a n and b n are

16 1.1 Basic Concepts and Formulae 3 a n = 1 L b n = 1 L L L L L f (x) cos(nπ x/l) dx (1.5) f (x)sin(nπ x/l) dx (1.6) Complex form of Fourier series Assuming that the Series (1.1) converges at f (x), f (x) = C ne inπx/l (1.7) n= with C n = 1 L C+2L C 1 f (x)e iπnx/l 2 (a n ib n ) n > 0 dx = 1 2 (a n + ib n ) n < a o n = 0 (1.8) Fourier transforms The Fourier transform of f (x) is defined as I( f (x)) = F(α) = f (x)e iαx dx (1.9) and the inverse Fourier transform of F(α)is I 1 ( f (α)) = F(x) = 1 F(α)e i x dα (1.10) 2π f (x) and F(α) are known as Fourier Transform pairs. Some selected pairs are given in Table 1.1. Table 1.1 f (x) F(α) f (x) F(α) 1 x 2 + a 2 x x 2 + a 2 1 x e ax a a α 2 + a 2 πiα a e aα e 1 π ax2 /4a 2 a e α2 π π xe ax2 /4a 2 4a 3/2 αe α2 πe aα Gamma and beta functions The gamma function Γ(n) is defined by

17 4 1 Mathematical Physics Γ(n) = 0 e x x n 1 dx (Re n > 0) (1.11) Γ(n + 1) = nγ(n) (1.12) If n is a positive integer Γ(n + 1) = n! (1.13) ( ) 1 Γ = ( ) ( ) 3 π 5 π; Γ = ; Γ = 3 π (1.14) 2 4 ( Γ n + 1 ) = (2n 1) π (n = 1, 2, 3,...) (1.15) 2 2 n ( Γ n + 1 ) = ( 1)n 2 n π (n = 1, 2, 3,...) (1.16) (2n 1) Γ(n + 1) = n! = 2πnn n e n (Stirling s formula) (1.17) n Beta function B(m, n) is defined as B(m, n) = Γ(m)Γ(n) Γ(m + n) (1.18) B(m, n) = B(n, m) (1.19) B(m, n) = 2 B(m, n) = π/2 0 0 sin 2m 1 θ cos 2n 1 θ dθ (1.20) t m 1 dt (1.21) (1 + t) m+n Special funtions, properties and differential equations Hermite functions: Differential equation: y 2xy + 2ny = 0 (1.22) when n = 0, 1, 2,...then we get Hermite s polynomials H n (x) ofdegreen, given by ( H n (x) = ( 1) n e x2 dn dx n e x2) (Rodrigue s formula)

18 1.1 Basic Concepts and Formulae 5 First few Hermite s polynomials are: H o (x) = 1, H 1 (x) = 2x, H 2 (x) = 4x 2 2 H 3 (x) = 8x 3 12x, H 4 (x) = 16x 4 48x (1.23) Generating function: e 2tx t 2 = n=0 H n (x)t n n! (1.24) Recurrence formulas: H n (x) = 2nH n 1(x) H n+1 (x) = 2xH n (x) 2nH n 1 (x) (1.25) Orthonormal properties: e x2 H m (x)h n (x) dx = 0 m n (1.26) e x2 {H n (x)} 2 dx = 2 n n! π (1.27) Legendre functions: Differential equation of order n: (1 x 2 )y 2xy + n(n + 1)y = 0 (1.28) when n = 0, 1, 2,...we get Legendre polynomials P n (x). First few polynomials are: d n P n (x) = 1 2 n n! dx (x 2 1) n (1.29) n Generating function: P o (x) = 1, P 1 (x) = x, P 2 (x) = 1 2 (3x 2 1) P 3 (x) = 1 2 (5x 3 3x), P 4 (x) = 1 8 (35x 4 30x 2 + 3) (1.30) 1 = P n(x)t n (1.31) 1 2tx + t 2 n=0

19 6 1 Mathematical Physics Recurrence formulas: Orthonormal properties: Other properties: xp n (x) P n 1 (x) = np n(x) P n+1 (x) P n 1 (x) = (2n + 1)P n(x) (1.32) P m (x)p n (x) dx = 0 m n (1.33) {P n (x)} 2 dx = 2 2n + 1 (1.34) P n (1) = 1, P n ( 1) = ( 1) n, P n ( x) = ( 1) n P n (x) (1.35) Associated Legendre functions: Differential equation: } (1 x 2 )y 2xy + {l(l + 1) m2 y = 0 (1.36) 1 x 2 Pl m (x) = (1 x 2 m/2 dm ) dx P l(x) (1.37) m where P l (x) are the Legendre polynomials stated previously, l being the positive integer. Orthonormal properties: P o l (x) = P l(x) (1.38) and P m l (x) = 0 ifm > n (1.39) P m n (x)pm l (x) dx = 0 n l (1.40) {Pl m (x)} 2 dx = 2 (l + m)! 2l + 1 (l m)! (1.41) Laguerre polynomials: Differential equation: xy + (1 x)y + ny = 0 (1.42) if n = 0, 1, 2,...we get Laguerre polynomials given by

20 1.1 Basic Concepts and Formulae 7 L n (x) = e x dn dx (x n e x ) n (Rodrigue s formula) (1.43) The first few polynomials are: L o (x) = 1, L 1 (x) = x + 1, L 2 (x) = x 2 4x + 2 L 3 (x) = x 3 + 9x 2 18x + 6, L 4 (x) = x 4 16x x 2 96x + 24 (1.44) Generating function: Recurrence formulas: e xs/(1 s) 1 s = n=0 L n (x)s n n! (1.45) L n+1 (x) (2n + 1 x)l n (x) + n 2 L n 1 (x) = 0 xl n (x) = nl n(x) n 2 L n 1 (x) (1.46) Orthonormal properties: 0 0 e x L m (x)l n (x) dx = 0 m n (1.47) e x {L n (x)} 2 dx = (n!) 2 (1.48) Bessel functions: (J n (x)) Differential equation of order n x 2 y + xy + (x 2 n 2 )y = 0 n 0 (1.49) Expansion formula: Properties: J n (x) = k=0 ( 1) k (x/2) 2k n k!γ(k + 1 n) (1.50) J n (x) = ( 1) n J n (x) n = 0, 1, 2,... (1.51) J o (x) = J 1(x) (1.52) J n+1 (x) = 2n x J n(x) J n 1 (x) (1.53) Generating function: e x(s 1/s)/2 = J n(x)t n (1.54) n=

21 8 1 Mathematical Physics Laplace transforms: Definition: A Laplace transform of the function F(t)is 0 F(t)e st dt = f (s) (1.55) The function f (s) is the Laplace transform of F(t). Symbolically, L{F(t)} = f (s) and F(t) = L 1 { f (s)} is the inverse Laplace transform of f (s). L 1 is called the inverse Laplace operator. Table of Laplace transforms: F(t) f (s) af 1 (t) + bf 2 (t) af 1 (s) + bf 2 (s) af(at) f (s/a) e at F(t) f (s a) F(t a) t > a 0 t < a e as f (s) 1 1 s 1 t t n 1 1 n = 1, 2, 3,... (n 1)! sn e at 1 sin at a cos at sinh at a cosh at s 2 s a 1 s 2 + a s 2 s 2 + a 2 1 s 2 a s 2 s 2 a 2 Calculus of variation The calculus of variation is concerned with the problem of finding a function y(x) such that a definite integral, taken over a function shall be a maximum or minimum. Let it be desired to find that function y(x) which will cause the integral I = x2 x 1 F(x, y, y )dx (1.56)

22 1.1 Basic Concepts and Formulae 9 to have a stationary value (maximum or minimum). The integrand is taken to be a function of the dependent variable y as well as the independent variable x and y = dy/dx. The limits x 1 and x 2 are fixed and at each of the limits y has definite value. The condition that I shall be stationary is given by Euler s equation F y d F = 0 (1.57) dx y When F does not depend explicitly on x, then a different form of the above equation is more useful F x d ( F y F ) = 0 (1.58) dx y which gives the result F y F y = Constant (1.59) Statistical distribution Binomial distribution The probability of obtaining x successes in N-independent trials of an event for which p is the probability of success and q the probability of failure in a single trial is given by the binomial distribution B(x). B(x) is normalized, i.e. B(x) = N! x!(n x)! px q N x = C N x px q N x (1.60) It is a discrete distribution. The mean value, The S.D., N x=0 B(x) = 1 (1.61) x =Np (1.62) σ = Npq (1.63)

23 10 1 Mathematical Physics Poisson distribution The probability that x events occur in unit time when the mean rate of occurrence is m, is given by the Poisson distribution P(x). P(x) = e m m x (x = 0, 1, 2,...) (1.64) x! The distribution P(x) is normalized, that is x=0 p(x) = 1 (1.65) This is also a discrete distribution. When NP is held fixed, the binomial distribution tends to Poisson distribution as N is increased to infinity. The expectation value, i.e. The S.D., Properties: x =m (1.66) σ = m (1.67) p m 1 = p m (1.68) p x 1 = x m p m and p x+1 = m m + 1 p x (1.69) Normal (Gaussian distribution) When p is held fixed, the binomial distribution tends to a Normal distribution as N is increased to infinity. It is a continuous distribution and has the form f (x) dx = 1 e (x m)2 /2σ 2 dx (1.70) 2πσ where m is the mean and σ is the S.D. The probability of the occurrence of a single random event in the interval m σ and m + σ is and that between m 2σ and m + 2σ is Interval distribution If the data contains N time intervals then the number of time intervals n between t 1 and t 2 is n = N(e at 1 e at 2 ) (1.71) where a is the average number of intervals per unit time. Short intervals are more favored than long intervals.

24 1.1 Basic Concepts and Formulae 11 Two limiting cases: (a) t 2 = ; N = N o e λt (Law of radioactivity) (1.72) This gives the number of surviving atoms at time t. (b) t 1 = 0; N = N o (1 e λt ) (1.73) For radioactive decays this gives the number of decays in time interval 0 and t. Above formulas are equally valid for length intervals such as interaction lengths. Moment generating function (MGF) MGF = Ee (x μ)t = E [1 + (x μ)t + (x μ) 2 t 2 ] 2! +... t 2 = μ 2 2! + μ t (1.74) 3! so that μ n,thenth moment about the mean is the coefficient of t n /n!. Propagation of errors If the error on the measurement of f (x, y,...)isσ f and that on x and y, σ x and σ y, respectively, and σ x and σ y are uncorrelated then σ 2 f = Thus, if f = x ± y, then σ f = ( σx 2 + σ ) y 2 1/2 And if f = x y then σ f f = ( ) f 2 ( ) f 2 σx 2 x + σy 2 + (1.75) y ( ) σ 2 x + σ 2 1/2 y x 2 y 2 Least square fit (a) Straight line: y = mx + c It is desired to fit pairs of points (x 1, y 1 ), (x 2, y 2 ),...,(x n, y n ) by a straight line Residue: S = n i=1 (y i mx i C) 2 Minimize the residue: s s = 0; m c = 0 The normal equations are: m n x 2 i=1 i + C n x i n x i y i = 0 i=1 i=1 m n x i + nc n y i = 0 i=1 i=1

25 12 1 Mathematical Physics which are to be solved as ordinary algebraic equations to determine the best values of m and C. (b) Parabola: y = a + bx + cx 2 Residue: S = n i=1 (y i a bx i cxi 2)2 Minimize the residue: s s s = 0; = 0; a b c = 0 The normal equations are: yi na b x i c xi 2 = 0 xi y i a x i b x 2 i c x 3 i = 0 x 2 i y i a x 2 i b x 3 i c x 4 i = 0 which are to be solved as ordinary algebraic equations to determine the best value of a, b and c. Numerical integration Since the value of a definite integral is a measure of the area under a curve, it follows that the accurate measurement of such an area will give the exact value of a definite integral; I = x 2 x 1 y(x)dx. The greater the number of intervals (i.e. the smaller Δx is), the closer will be the sum of the areas under consideration. Trapezoidal rule area = ( 1 2 y 0 + y 1 + y 2 + y n ) 2 y n Δx (1.76) Simpson s rule area = Δx 3 (y 0 + 4y 1 + 2y 2 + 4y 3 + 2y 4 + y n ), nbeingeven. (1.77) Fig. 1.1 Integration by Simpson s rule and Trapezoidal rule

26 1.1 Basic Concepts and Formulae 13 Matrices Types of matrices and definitions Identity matrix: I 2 = ( ) 10 ; I 01 3 = (1.78) 001 Scalar matrix: ( ) a a ; 0 a 0 a 22 0 (1.79) a 33 Symmetric matrix: ( ) a 11 a 12 a 13 a ji = a ij ; a 12 a 22 a 23 (1.80) a 13 a 23 a 33 Skew symmetric: ( ) a 11 a 12 a 13 a ji = a ij ; a 12 a 22 a 23 (1.81) a 13 a 23 a 33 The Inverse of a matrix B = A 1 (B equals A inverse): if AB = BA = I and further, (AB) 1 = B 1 A 1 A commutes with B if AB = BA A anti-commutes with B if AB = B A The Transpose (A ) of a matrix A means interchanging rows and columns. Further, (A + B) = A + B (A ) = A, (ka) = ka (1.82) The Conjugate of a matrix. If a matrix has complex numbers as elements, and if each number is replaced by its conjugate, then the new matrix is called the conjugate and denoted by A or A (A conjugate) The Trace (Tr) or Spur of a matix is the num of the diagonal elements. Tr = a ii (1.83)

27 14 1 Mathematical Physics Hermetian matrix If A = A, so that a ij = a ji for all values of i and j. Diagonal elements of an Hermitian matrix are real numbers. Orthogonal matrix A square matrix is said to be orthogonal if AA = A A = I, i.e. A = A 1 The column vector (row vectors) of an orthogonal matrix A are mutually orthogonal unit vectors. The inverse and the transpose of an orthogonal matrix are orthogonal. The product of any two or more orthogonal matrices is orthogonal. The determinant of an Orthogonal matrix is ±1. Unitary matrix A square matrix is called a unitary matrix if (A) A = A(A) = I, i.e. if (A) = A 1. The column vectors (row vectors) of an n-square unitary matrix are an orthonormal set. The inverse and the transpose of a unitary matrix are unitary. The product of two or more unitary matrices is unitary. The determinant of a unitary matrix has absolute value 1. Unitary transformations The linear transformation Y = AX (where A is unitary and X is a vector), is called a unitary transformation. If the matrix is unitary, the linear transformation preserves length. Rank of a matrix If A 0, it is called non-singular; if A =0, it is called singular. A non-singular matrix is said to have rank r if at least one of its r-square minors is non-zero while if every (r + 1) minor, if it exists, is zero. Elementary transformations (i) The interchange of the ith rows and jth rows or ith column or jth column. (ii) The multiplication of every element of the ith row or ith column by a non-zero scalar. (iii) The addition to the elements of the ith row (column) by k (a scalar) times the corresponding elements of the jth row (column). These elementary transformations known as row elementary or row transformations do not change the order of the matrix.

28 1.1 Basic Concepts and Formulae 15 Equivalence A and B are said to be equivalent (A B) if one can be obtained from the other by a sequence of elementary transformations. The adjoint of a square matrix If A = [a ij ] is a square matrix and α ij the cofactor of a ij then α 11 α 21 α n1 adj A = α 12 α 22 α nn The cofactor α ij = ( 1) i+ j M ij where M ij is the minor obtained by striking off the ith row and jth column and computing the determinant from the remaining elements. Inverse from the adjoint Inverse for orthogonal matrices A 1 = adj A A Inverse of unitary matrices A 1 = A Characteristic equation A 1 = (A) Let AX = λx (1.84) be the transformation of the vector X into λx, where λ is a number, then λ is called the eigen or characteristic value. From (1.84): (A λi )X = a 11 λ a 12 a 1n a 21 a 22 λ a 2n. a n1 a nn λ. x 1 x 2. x n = 0 (1.85)

29 16 1 Mathematical Physics The system of homogenous equations has non-trivial solutions if a 11 λ a 12 a 1n a 21 a 22 λ a 2n A λi =. = 0 (1.86) a n1 a nn λ The expansion of this determinant yields a polynomial φ(λ) = 0 is called the characteristic equation of A and its roots λ 1,λ 1,...,λ n are known as the characteristic roots of A. The vectors associated with the characteristic roots are called invariant or characteristic vectors. Diagonalization of a square matrix If a matrix C is found such that the matrix A is diagonalized to S by the transformation S = C 1 AC (1.87) then S will have the characteristic roots as the diagonal elements. Ordinary differential equations The methods of solving typical ordinary differential equations are from the book Differential and Integral Calculus by William A. Granville published by Ginn & Co., An ordinary differential equation involves only one independent variable, while a partial differential equation involves more than one independent variable. The order of a differential equation is that of the highest derivative in it. The degree of a differential equation which is algebraic in the derivatives is the power of the highest derivative in it when the equation is free from radicals and fractions. Differential equations of the first order and of the first degree Such an equation must be brought into the form Mdx+Ndy = 0, in which M and N are functions of x and y. Type I variables separable When the terms of a differential equation can be so arranged that it takes on the form (A) f (x) dx + F(y)dy = 0 where f (x) is a function of x alone and F(y) is a function of y alone, the process is called separation of variables and the solution is obtained by direct integration. (B) f (x) dx + F(y)dy = C where C is an arbitrary constant.

30 1.1 Basic Concepts and Formulae 17 Equations which are not in the simple form (A) can be brought into that form by the following rule for separating the variables. First step: Clear off fractions, and if the equation involves derivatives, multiply through by the differential of the independent variable. Second step: Collect all the terms containing the same differential into a single term. If then the equation takes on the form XY dx + X Y dy = 0 where X, X are functions of x alone, and Y, Y are functions of y alone, it may be brought to the form (A) by dividing through by X Y. Third step: Integrate each part separately as in (B). Type II homogeneous equations The differential equation Mdx + Ndy = 0 is said to be homogeneous when M and N are homogeneous function of x and y of the same degree. In effect a function of x and y is said to be homogenous in the variable if the result of replacing x and y by λx and λy (λ being arbitrary) reduces to the original function multiplied by some power of λ. This power of λ is called the degree of the original function. Such differential equations may be solved by making the substitution y = vx This will give a differential equation in v and x in which the variables are separable, and hence we may follow the rule (A) of type I. Type III linear equations A differential equation is said to be linear if the equation is of the first degree in the dependent variables (usually y) and its derivatives. The linear differential equation of the first order is of the form dy dx + Py = Q where P, Q are functions of x alone, or constants, the solution is given by ye Pdx = Qe Pdx dx + C